Lesson 7: Part 1 of Cluster and Systematic Sampling

Lesson 7: Part 1 of Cluster and Systematic Sampling

Overview

In Section 7.1, we introduce cluster and systematic sampling and show their similar structure. Graphical representations of primary units and secondary units are given. Notations are introduced.

In Section 7.2, when primary units are selected by srs, unbiased estimators and ratio estimators for cluster sampling are provided. Basic principles to obtain estimators of low variances are discussed. Then we discuss why and when will we use cluster sampling. That is followed by an example showing how to compute the ratio estimator and the unbiased estimator when the cluster sampling with primary units selected by srs is used.

In Section 7.3, cluster sampling with primary units selected by probabilities proportional to size is discussed. Then an example is given.

 Lesson 7: Ch. 12.1-12.3 of Sampling by Steven Thompson, 3rd edition

Objectives

Upon completion of this lesson, you should be able to:

  • know why and when to use cluster sampling,
  • know the notation for cluster and systematic sampling,
  • know what are primary units and what are secondary units,
  • compute the unbiased estimator for cluster samples when primary units are selected by srs,
  • compute the ratio estimator for cluster samples when primary units are selected by srs, and
  • compute the Hansen-Hurwitz estimator for cluster samples when primary units are selected by pps.

7.1 - Introduction to Cluster and Systematic Sampling

7.1 - Introduction to Cluster and Systematic Sampling

On the surface, systematic and cluster sampling are very different. In fact, the two designs share the same structure: the population is partitioned into primary units, each primary unit being composed of secondary units. Whenever a primary unit is included in the sample, the y-values of every secondary unit within it are observed.

Example: An one in three systematic sampling where we randomly pick one from the first three units and then choose every three from that on.

 

figure 12.1

Randomly pick a value from {1, 2, 3}. For example, if 2 is chosen, then we will pick {2, 5, 8, 11, 14}, the figure 12.1's. The set {2, 5, 8, 11, 14} is an example of a primary unit.

It is not uncommon to have a systematic sample of size 1, such as the above 1 in 3 systematic sample. We just sample 1 primary unit.

In the following two graphs, we provide examples for two configurations of primary units:

figure 12.1

The above figure has 50 primary units (PSU)
(the colored rectangle is an example of a primary unit of cluster sampling)

figure 12.2

The above figure has 25 primary units (PSU)
(the colored units (collectively) is an example of a primary unit of a systematic sampling)

Primary units (PSU) may be different from observation units. One can view the systematic sampling as a sampling of primary units. Once the primary units are selected, a cluster of secondary units are also selected.

Advantages of Systematic Sampling

  1. Easier to perform in the field, especially if a good frame is not available.
  2. Frequently provides more information per unit cost than simple random sampling, in the sense of smaller variances.

For example, a systematic sample was drawn from a batch of produced computer chips. The first 400 chips are fine but due to a fault of the machine, the last 300 chips are defective. Systematic sampling will select uniformly over the defective and non-defective items and would give a very accurate estimate of the fraction of defective items.

Cluster Sampling and Systematic Sampling

A cluster/systematic sample is a probability sample in which each sampling unit is a collection, or cluster, of elements.

Notations of cluster and systematic sampling:
  • N : the number of primary units in the population n : the number of primary units in the sample \(M_i\): the number of secondary units in the ith primary unit
  • \(M=\sum\limits_{i=1}^N M_i\): the total number of secondary units in the population
  • \(y_{ij}\): the value of the variable of interest of jth secondary unit in the ith primary unit
  • \(y_i=\sum\limits_{j=1}^{M_i}y_{ij}\): the total (i.e. sum) of y-values in the ith primary unit

For Fig. 1 below, N = 50, n = 10, \(M_i\) = 8

figure 12.1
Fig. 1

For Fig. 2 below, N = 25, n = 2, \(M_i\) = 16

figure 12.2
Fig. 2

Figure 1 shows an example of cluster sampling and figure 2 shows an example of systematic sampling. Secondary units of a primary unit of the cluster sampling are close together whereas secondary units of a primary unit of the systematic sampling separate from each other.

More Notation:

Thus, the population total is:

\(\tau=\sum\limits_{i=1}^N \sum\limits_{j=1}^{M_i}y_{ij}=\sum\limits_{i=1}^N y_i\)

The population mean per primary unit is:

\(\mu_1=\tau/N\)

The population mean per secondary unit is

\(\mu=\tau/M\).


7.2 - Estimators for Cluster Sampling when Primary units are selected by simple random sampling

7.2 - Estimators for Cluster Sampling when Primary units are selected by simple random sampling

When the primary units are selected by simple random sampling, frequently used estimators among many possible estimators are:

A. Unbiased estimator

\(\hat{\tau}=N\cdot \bar{y}=\dfrac{N\cdot \sum\limits_{i=1}^n y_i}{n}\)

Recall that yi is the total of y-values in the ith primary unit.

\(\hat{V}ar(\hat{\tau})=N\cdot (N-n)\dfrac{s^2_u}{n}\)

where \(s^2_u=\dfrac{1}{n-1}\sum\limits_{i=1}^n(y_i-\bar{y})^2\)

To estimate the mean per primary unit,\(\tau\)/ N, the mean and variance equations are given below:

\(\bar{y}=\dfrac{\hat{\tau}}{N}\), \(\hat{Var}(\bar{y})=\dfrac{1}{N^2} \hat{Var}(\hat{\tau})\)

To estimate the mean per secondary unit, the mean and variance equations are given below:

\(\hat{\mu}=\dfrac{\hat{\tau}}{M}\), \(\hat{Var}(\hat{\mu})=\dfrac{1}{M^2} \hat{Var}(\hat{\tau})\)

B. Ratio Estimator

If the primary unit total  yi is highly correlated with the primary unit size Mi , a ratio estimator based on size may be efficient. 

\(\hat{\tau}_r=r \cdot M,\quad M=\sum\limits_{i=1}^N M_i\)

where \(r=\dfrac{\sum\limits_{i=1}^n y_i}{\sum\limits_{i=1}^n M_i},\quad \hat{V}ar(\hat{\tau}_r)=\dfrac{N(N-n)}{n(n-1)}\sum\limits_{i=1}^n (y_i-rM_i)^2\)

The Basic Principle

Since every secondary unit is observed within a selected primary unit, the within primary unit variance does not enter into the variances of the estimators. For example,

\(\hat{V}ar(\hat{\tau})=N(N-n)\cdot \dfrac{s^2_u}{n}\)
where  \(s^2_u=\dfrac{1}{n-1}\sum\limits_{i=1}^n (y_i-\bar{y})^2\)

Thus, to obtain estimators of low variances,

  1. Clusters should be formed so that one cluster is similar to another cluster. (Note: this is 'very different' from saying that units in the cluster are similar)
  2. Each cluster should contain the full diversity of the population and thus, is 'representative'.

With natural populations of spatially distributed plants, animals, or minerals, and human populations, the above condition is typically satisfied by systematic sampling where each cluster contains units that are far apart. Cluster sampling is more often than not carried out for reasons of convenience or practicality rather than to obtain the lowest variances.

Why or When do we use cluster sampling?

Will it give us a more precise estimator? The answer is no for most cases.

We do use cluster sampling out of necessity even though it will give us a larger variance.

If the objective of sampling is to obtain a specified amount of information about a population parameter at minimum cost, cluster sampling sometimes gives more information per unit cost than simple random sampling, stratified sampling and systematic sampling due to the cost of sampling units within a cluster may be much lower.

Cluster sampling is an effective design in two different scenarios:

  1. A good frame listing the population elements either is not available or is very costly to obtain, whereas a frame listing clusters is easily obtained.
  2. The cost of obtaining observations increases as the distance separating the elements increases.

Example 7-1: Average yearly Vacation Budget

Let's look at an example of cluster sampling using a ratio estimator.

A sociologist wants to estimate the average yearly vacation budget for each household in a certain city. It is given that there are 3,100 households in the city. The sociologist marked off the city into 400 blocks and treated them as 400 clusters. He then randomly sampled 24 clusters interviewing every household living in that cluster. The data are given in the table below:

Cluster Number of households \(M_i\) Total vacation budget per cluster \(y_i\)
1 7 12,000
2 9 15,000
3 5 8,000
4 8 13,000
5 12 18,000
6 5 7,000
7 4 6,000
8 8 13,000
9 14 22,000
10 6 9,800
11 3 7,000
12 13 18,000
13 8 12,340
14 4 5,000
15 6 8,900
16 9 14,000
17 3 4,000
18 10 11,400
19 4 5,000
20 7 13,000
21 6 8,900
22 5 8,700
23 7 10,000
24 6 9,200
  169 259,240

  Using Minitab

To use Minitab to plot total for cluster versus cluster size:

  1. Graph > Scatterplot
  2. Select 'Total for Cluster' as the Y variable
  3. Select 'Cluster Size' as the X variable

To use Minitab to display descriptive statistics:

Stat > Basic Statistics > Display Descriptive Statistics

Here is a plot of this data so that we can see if the cluster size is proportional to the total for the cluster.

Minitab output

Minitab output of descriptive statistics:

Minitab output

The regression equation is

total for the cluster = 648 + 1442 cluster size

Coefficients
Predictor Coef StDev T P
Constant 648.0 705.9 0.92 0.369
cluster 1441.94 92.59 15.57 0.000
Descriptive Statistics for the variables:
Variable N Mean StDev
Cluster size 24 7.042 2.985
\(Y_i\) 24 10802 4495
\Y_i -rM_i\) 24 -0 1325

The ratio estimator for cluster sample (ratio-to-size):

If primary unit total \(y_i\) is highly correlated with cluster size \(M_i\), a ratio estimator based on size may be efficient. The ratio estimator of the population total is:

 \(\hat{\tau}_r=r\cdot M \quad \text{where } r=\dfrac{\sum\limits_{i=1}^n y_i}{\sum\limits_{i=1}^n M_i}\)

The ratio estimator is biased but the bias is small when the sample size is large. Here is the variance:

\(\hat{V}ar(\hat{\tau}_r)=\dfrac{N(N-n)}{n(n-1)}\sum\limits_{i=1}^n (y_i-rM_i)^2\)

To estimate the population mean per secondary unit we have: \(\mu\) =\(\tau\)/ M

The ratio estimator is:

\(\hat{\mu}_r=\dfrac{\hat{\tau}_r}{M}=r\)

\(\hat{V}ar(\hat{\mu}_r)=\dfrac{N(N-n)}{n(n-1)}\cdot \dfrac{1}{M^2} \sum\limits_{i=1}^n (y_i-rM_i)^2\)

Back to the example. To estimate the average yearly vacation budget for each household we will use:

\(\hat{\mu}_r=r=\dfrac{\sum\limits_{i=1}^n y_i}{\sum\limits_{i=1}^n M_i}\)

In this example we see that N = 400, the total number of blocks, and n = 24. M in this case is as follows:

\(M=\sum\limits_{i=1}^N M_i=3100\)

Try it!

Find the ratio estimator for the average yearly vacation budget for each household in that city. Also, find the estimated variance for the ratio estimator.

\(\hat{\mu}_r=\dfrac{\sum\limits_{i=1}^n y_i}{\sum\limits_{i=1}^n M_i}=\dfrac{259240}{169}=1533.96\)

\(\hat{V}ar(\hat{\mu}_r)=\dfrac{N(N-n)}{n\cdot M^2}\cdot \dfrac{1}{n-1} \sum\limits_{i=1}^n (y_i-rM_i)^2\)

For this example, M = 3100, N = 400, n = 24

\begin{align}
\dfrac{1}{n-1}\sum\limits_{i=1}^n (y_i-rM_i)^2 &=[\text{st.dev. of }(y-rM)]^2\\
&= (1325)^2\\
\end{align}

\begin{align}
\hat{V}ar(\hat{\mu}_r)&=\dfrac{400(400-24)}{24(3100)^2}\cdot (1325)^2\\
&= 1144.84\\
\end{align}

 If we used the unbiased estimator would our variance be larger or smaller?

For this example, we also want to compute the unbiased estimator for comparison purposes.

Try it!

Find the unbiased estimator for the average yearly vacation budget for each household in that city. Also, find the estimated variance for the unbiased estimator.

\begin{align}
\hat{\mu}&= N \dfrac{\sum\limits_{i=1}^n y_i }{n}\cdot \dfrac{1}{M}\\
&= 400 \cdot \dfrac{259240}{24}\cdot \dfrac{1}{3100}\\
&= 400 \cdot 10802 \cdot \dfrac{1}{3100}\\
&= 1393.81\\
\end{align}

\begin{align}
\hat{V}ar(\hat{\mu})&=\dfrac{N(N-n)}{M^2 \cdot n} \cdot \dfrac{1}{n-1}\sum\limits_{i=1}^n (y_i-\bar{y})^2\\
&= \dfrac{400(400-24)}{(3100)^2 \cdot 24}(\text{st.dev. of }y)^2\\
&= \dfrac{400(400-24)}{(3100)^2 \cdot 24}(4495)^2\\
&= 13175.67\\
\end{align}

Remark 1: This variance is huge and we should be very unhappy using the unbiased estimate. We can thus see that when cluster total is proportional to cluster size, it is better to use the ratio estimate than the unbiased estimator.
Remark 2: Can we use a formula to compute variances by the simple random sampling? Unfortunately, No! We would have to have collected this data via simple random sampling in order to calculate the variance by the formula corresponding to simple random sampling. Note: it is a big mistake if you do not compute the variance according to its sampling scheme!

Using R

Here is the code for R for this example:

Datafile: Vacation.txt
R code: Chapter7_Vacation Budget.R.txt


7.3 - Estimator for Cluster Sampling when Primary units are selected by p.p.s

7.3 - Estimator for Cluster Sampling when Primary units are selected by p.p.s

The primary units selected with probabilities proportional to size:

\(p_i=M_i/M\)

The Hansen-Hurwitz (p.p.s.) estimator is:

\(\hat{\tau}_p=\dfrac{M}{n}\sum\limits_{i=1}^n \left(\dfrac{y_i}{M_i}\right)\)

Denote by \(\bar{y}_i=\dfrac{y_i}{M_i}\)

\(\hat{V}ar(\hat{\tau}_p)=\dfrac{M^2}{n(n-1)}\sum\limits_{i=1}^n (\bar{y}_i-\hat{\mu}_p)^2\) where

\(\hat{\mu}_p=\dfrac{\hat{\tau}_p}{M}\) is unbiased for \(\mu\).

Thus we also see that:

\(\hat{V}ar(\hat{\mu}_p)=\dfrac{1}{n(n-1)}\sum\limits_{i=1}^n (\bar{y}_i-\hat{\mu}_p)^2\)

Example: Estimating population mean per secondary unit when primary units are selected by pps

From the "Total number of computer help requests" example in Lesson 3.1, 3 clusters out of 10 clusters are sampled (n = 3) with replacement. The data are:

\(y_1=420, y_2 = 1785, y_3=2198\)

\(M_1=650, M_2=2840, M_3=3200\)

Try it!

Find the Hansen-Hurwitz estimator for the population mean and also find the variance of the estimator.

 \begin{align}
\hat{\mu}_p &= \dfrac{1}{n} \sum\limits_{i=1}^n \dfrac{y_i}{M_i}\\
&= \dfrac{1}{3}\times \left(\dfrac{420}{650}+\dfrac{1785}{2840}+\dfrac{2198}{3200}\right)\\
&= 0.6538\\
\end{align}

\begin{align}
\hat{V}ar(\hat{\mu}_p)&=\dfrac{1}{n(n-1)}\sum\limits_{i=1}^n (\bar{y}_i-\hat{\mu}_p)^2\\
&= \dfrac{1}{3 \times 2}[(0.6462-0.6538)^2+(0.6285-0.6538)^2+(0.6869-0.6538)^2]\\
&= 0.000299\\
\end{align}

Note! For an example to review an estimate the population total, refer to earlier lecture notes on the Hansen-Hurwitz estimator and the probabilities proportional to size as they were referred to in the Palm Tree total estimator examples.

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